To answer this question, we investigate \(\tilde{T}_c\)  as a function of \(|\tilde{\varepsilon }_0|\)  at the fixed chemical potential \(\tilde{\mu }=0.3\)  (near the maximum of \(\tilde{T}_{c0}\) ). In particular, Fig. REF  presents \(\tilde{T}_c\)  versus \(|\tilde{\varepsilon }_0|\)  for the three different values of the pair-exchange coupling \(\lambda _{12}=0.005, 0.01\) , and \(0.02\)  at \(m_2/m_1=1\) . In agreement with Fig. REF , \(\tilde{T}_c\)  increases with \(|\tilde{\varepsilon }_0|\)  and at sufficiently large values of \(|\tilde{\varepsilon }_0|\)  the fluctuation-shifted critical temperature approaches the mean-field superconducting temperature \(\tilde{T}_{c0}\approx 0.155\) . The latter is almost independent of \(\lambda _{12}\)  as the mean-field temperature is mainly governed by the Q1D Feshbach-like resonance [1]}. At small \(|\tilde{\varepsilon }_0|\)  we find that \(\tilde{T}_c\)  deviates significantly from \(\tilde{T}_{c0}\) . However, surprisingly, a sharp drop of \(\tilde{T}_c\)  occurs only in the region \(|\tilde{\varepsilon }_0| \lesssim 10\) . In particular, one finds that the fluctuation-shifted critical temperature can still be significant (e.g., larger than a half of \(\tilde{T}_{c0}\) ) for a nearly shallow band 1, where \(|\tilde{\varepsilon }_0| \approx 0.5\) -\(1.0\) , as is seen for \(\lambda _{12}=0.02\) . Notice that in this case the relative Fermi energy \(\tilde{E}_F=|\tilde{\varepsilon }_0|+\tilde{\mu }\)  (in units of \(\hbar \omega _c\) ) is about 1-2 and thus, we arrive at the estimate \(E_F/T_c \approx 10\) -20, which is orders of magnitude smaller than the corresponding values in conventional superconductors.
